There are many instances where it is desired to mount safety apparatus or construction apparatus or other type of apparatus to a beam or a rail. Some examples of mounting apparatus for this purpose are found in U.S. Pat. Nos. 5,526,896; 2,765,139; 5,354,026; 4,767,091 and 4,799,639. This application discloses clamping apparatus that functions to provide a strong support for the mounting of any desired type of apparatus to beams or rails of varying configurations.